Why is mesh reacting weirdly when adding a Solidify modifier or smooth shading?
yes you must have made some bad manipulations but it's hard to tell how it happened. Anyway, you can select all in Edit mode and W > Remove Doubles, with a Merge Distance of 0.02 on the bottom of the left (Tools) panel. Also, recalculate the normals with a ctrlN. Then some isolate vertices will remain and you can delete them: select the main shape with L and invert the selection with ctrli, then X to delete. At last, you have some tris that you could easily replace with quads.